Average Error: 16.4 → 0.2
Time: 5.9s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999997606884:\\ \;\;\;\;\left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right) - \left(\frac{2}{\alpha \cdot \alpha} + \mathsf{fma}\left(3, \frac{\beta}{\alpha \cdot \alpha}, \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{\alpha - \beta}{\beta + \left(\alpha + 2\right)}, -0.5, 0.5\right)\\ \mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(e^{{t_0}^{0.6666666666666666}}\right)\right)\right) \cdot \sqrt[3]{t_0} \end{array}\\ \end{array} \]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999997606884:\\
\;\;\;\;\left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right) - \left(\frac{2}{\alpha \cdot \alpha} + \mathsf{fma}\left(3, \frac{\beta}{\alpha \cdot \alpha}, \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := \mathsf{fma}\left(\frac{\alpha - \beta}{\beta + \left(\alpha + 2\right)}, -0.5, 0.5\right)\\
\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(e^{{t_0}^{0.6666666666666666}}\right)\right)\right) \cdot \sqrt[3]{t_0}
\end{array}\\


\end{array}

(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))
↓
(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999999997606884)
   (-
    (+ (/ 1.0 alpha) (/ beta alpha))
    (+
     (/ 2.0 (* alpha alpha))
     (fma 3.0 (/ beta (* alpha alpha)) (* (/ beta alpha) (/ beta alpha)))))
   (let* ((t_0 (fma (/ (- alpha beta) (+ beta (+ alpha 2.0))) -0.5 0.5)))
     (* (log1p (expm1 (log (exp (pow t_0 0.6666666666666666))))) (cbrt t_0)))))
double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}
↓
double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999997606884) {
		tmp = ((1.0 / alpha) + (beta / alpha)) - ((2.0 / (alpha * alpha)) + fma(3.0, (beta / (alpha * alpha)), ((beta / alpha) * (beta / alpha))));
	} else {
		double t_0 = fma(((alpha - beta) / (beta + (alpha + 2.0))), -0.5, 0.5);
		tmp = log1p(expm1(log(exp(pow(t_0, 0.6666666666666666))))) * cbrt(t_0);
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999999999760688429

    1. Initial program 60.0

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]

    2. Simplified60.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}, -0.5, 0.5\right)} \]

    3. Taylor expanded around inf 2.9

      \[\leadsto \color{blue}{\left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right) - \left(2 \cdot \frac{1}{{\alpha}^{2}} + \left(3 \cdot \frac{\beta}{{\alpha}^{2}} + \frac{{\beta}^{2}}{{\alpha}^{2}}\right)\right)} \]

    4. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right) - \left(\frac{2}{\alpha \cdot \alpha} + \mathsf{fma}\left(3, \frac{\beta}{\alpha \cdot \alpha}, \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)} \]

    if -0.999999999760688429 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

    1. Initial program 0.2

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]

    2. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}, -0.5, 0.5\right)} \]
    3. Using strategy rm

    4. Applied add-cube-cbrt_binary640.8

      \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}, -0.5, 0.5\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}, -0.5, 0.5\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}, -0.5, 0.5\right)}} \]

    5. Simplified0.8

      \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\frac{\alpha - \beta}{\beta + \left(2 + \alpha\right)}, -0.5, 0.5\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\alpha - \beta}{\beta + \left(2 + \alpha\right)}, -0.5, 0.5\right)}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}, -0.5, 0.5\right)} \]

    6. Simplified0.8

      \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(\frac{\alpha - \beta}{\beta + \left(2 + \alpha\right)}, -0.5, 0.5\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\alpha - \beta}{\beta + \left(2 + \alpha\right)}, -0.5, 0.5\right)}\right) \cdot \color{blue}{\sqrt[3]{\mathsf{fma}\left(\frac{\alpha - \beta}{\beta + \left(2 + \alpha\right)}, -0.5, 0.5\right)}} \]
    7. Using strategy rm

    8. Applied add-log-exp_binary640.8

      \[\leadsto \color{blue}{\log \left(e^{\sqrt[3]{\mathsf{fma}\left(\frac{\alpha - \beta}{\beta + \left(2 + \alpha\right)}, -0.5, 0.5\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\alpha - \beta}{\beta + \left(2 + \alpha\right)}, -0.5, 0.5\right)}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\alpha - \beta}{\beta + \left(2 + \alpha\right)}, -0.5, 0.5\right)} \]

    9. Simplified0.2

      \[\leadsto \log \color{blue}{\left(e^{{\left(\mathsf{fma}\left(\frac{\alpha - \beta}{\beta + \left(2 + \alpha\right)}, -0.5, 0.5\right)\right)}^{0.6666666666666666}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\alpha - \beta}{\beta + \left(2 + \alpha\right)}, -0.5, 0.5\right)} \]
    10. Using strategy rm

    11. Applied log1p-expm1-u_binary640.2

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(e^{{\left(\mathsf{fma}\left(\frac{\alpha - \beta}{\beta + \left(2 + \alpha\right)}, -0.5, 0.5\right)\right)}^{0.6666666666666666}}\right)\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\alpha - \beta}{\beta + \left(2 + \alpha\right)}, -0.5, 0.5\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999997606884:\\ \;\;\;\;\left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right) - \left(\frac{2}{\alpha \cdot \alpha} + \mathsf{fma}\left(3, \frac{\beta}{\alpha \cdot \alpha}, \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(e^{{\left(\mathsf{fma}\left(\frac{\alpha - \beta}{\beta + \left(\alpha + 2\right)}, -0.5, 0.5\right)\right)}^{0.6666666666666666}}\right)\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\alpha - \beta}{\beta + \left(\alpha + 2\right)}, -0.5, 0.5\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))